Exact Algorithms for Weak Roman Domination

Abstract

International audienceWe consider the Weak Roman Domination problem. Given an undirected graph G = (V,E), the aim is to find a weak roman domination function (wrd-function for short) of minimum cost, i.e. a function f: V → {0,1,2} such that every vertex v ∈ V is defended (i.e. there exists a neighbor u of v, possibly u = v, such that f(u)≥1) and for every vertex v ∈ V with f(v) = 0 there exists a neighbor u of v such that f(u)≥1 and the function f_{u → v} defined by:f_u→v(x)={1 if x=v,f(u)-1 if x=u,f(x) if x∉{u,v}} does not contain any undefended vertex. The cost of a wrd-function f is defined by cost(f) = ∑ _{v ∈ V} f(v). The trivial enumeration algorithm runs in time O^∗(3ⁿ) and polynomial space and is the best one known for the problem so far. We are breaking the trivial enumeration barrier by providing two faster algorithms: we first prove that the problem can be solved in O^∗(2ⁿ) time needing exponential space, and then describe an O^∗(2.2279ⁿ) algorithm using polynomial space. Our results rely on structural properties of a wrd-function, as well as on the best polynomial space algorithm for the Red-Blue Dominating Set problem

Minimal Roman Dominating Functions: Extensions and Enumeration

“…Roman Domination has received a lot of attention from the algorithmic community in the past 15 years [2,[23][24][25][26][27][28][29][30][31]. Relevant to our paper is the development of exact algorithms for Roman Domination: combining ideas from [26,32], an O(1.5014 n ) exponential-time and -space algorithm (making use of known Set Cover algorithms via a transformation to Partial Dominating Set) was presented in [33].…”

Roman Census: Enumerating and Counting Roman Dominating Functions on Graph Classes

“…It is easy to view Roman Domination as a graph-theoretic problem, where the map is modeled as a graph. Roman Domination has received notable attention in the last two decades [7,17,23,26,40,41,44,48,49,51]. Relevant to our work is the development of exact algorithms: Roman Domination can be solved in O(1.5014 n ) time (and space), see [40,52,54].…”

Minimal Roman Dominating Functions: Extensions and Enumeration

“…Roman Domination has received a lot of attention from the algorithmic community in the past 15 years [4,15,21,24,35,36,39,43,44,47]. Relevant to our paper is the development of exact algorithms for Roman Domination: combining ideas from [35,46], an O(1.5014 n ) exponential-time and -space algorithm (making use of known Set Cover algorithms via a transformation to Partial Dominating Set) was presented in [48].…”